Cartan-covariant quantum channels are introduced and studied using the Choi-Jamiołkowski isomorphism. The channels are Cartan-covariant as the covariance groups considered form symmetric pairs with the special unitary group SU( $n$ ), and their associated Cartan involution provides a route to exactly compute the eigenspectrum of their Choi states and the partial transpose for any $n\in \mathbb{N}$ . These channels include previously studied SO $(n)$ -covariant channels, and further include Sp $(\frac{n}{2})$ -covariant channels (when $n$ is even) and S(U( $p$ ) $\times$ U( $q$ ))-covariant channels (where $n\!=\!p\!+\!q$ ). We show that all Cartan-covariant channels satisfy the PPT $^{2}$ -conjecture and further demonstrate the nontrivial nature of this result for the class of Sp $(\frac{n}{2})$ -covariant and ${\rm S}({\rm U}(p) \!\times\!{\rm U}(q))$ -covariant channelsCartan-covariant quantum channels are introduced and studied using the Choi-Jamiołkowski isomorphism. The channels are Cartan-covariant as the covariance groups considered form symmetric pairs with the special unitary group SU($n$), and their associated Cartan involution provides a route to exactly compute the eigenspectrum of their Choi states and the partial transpose for any $n\in \mathbb{N}$. These channels include previously studied SO$(n)$-covariant channels, and further include Sp$(\frac{n}{2})$-covariant channels (when $n$ is even) and S(U($p$) $\times$ U($q$))-covariant channels (where $n\!=\!p\!+\!q$). We show that all Cartan-covariant channels satisfy the PPT$^{2}$-conjecture and further demonstrate the nontrivial nature of this result for the class of Sp$(\frac{n}{2})$-covariant and ${\rm S}({\rm U}(p) \!\times\!{\rm U}(q))$-covariant channels

arXiv:2501.03959Jan 7, 20250 citations

Invariance

Klaus Mainzer

Compendium of Quantum PhysicsJan 1, 20090 citations

Deriving the generalized momentum operators from covariant derivatives

Mehdi Jafari Matehkolaee

The generalized momentum operators are derived in the framework of non-relativistic quantum mechanics, taking into account an especial assumption on the covariant derivative. According to this assumption, a scalar density doesn’t change under the action of covariant derivative. The Hermitian form of the covariant derivative is discussed. It is shown that, with the help of the adjoint of covariant derivatives, generalized momentum operators can be derived. The inverse of covariant derivative is calculated. It is shown that the inverse of generalized momentum operators can be deduced from the inverse of covariant derivatives.The generalized momentum operators are derived in the framework of non-relativistic quantum mechanics, taking into account an especial assumption on the covariant derivative. According to this assumption, a scalar density doesn’t change under the action of covariant derivative. The Hermitian form of the covariant derivative is discussed. It is shown that, with the help of the adjoint of covariant derivatives, generalized momentum operators can be derived. The inverse of covariant derivative is calculated. It is shown that the inverse of generalized momentum operators can be deduced from the inverse of covariant derivatives.

International Journal of Geometric Methods in Modern PhysicsJul 20, 20231 citation

Scale Invariance

Annick Lesne, Michel Laguës

Springer eBooksNov 2, 201135 citations

Covariance Geometry of Basis-Resolved Low-Energy Wavefunctions in the Spin- $1/2$ Kitaev--Heisenberg ModelCovariance Geometry of Basis-Resolved Low-Energy Wavefunctions in the Spin-$1/2$ Kitaev--Heisenberg Model

Sk Saniur Rahaman, Syed R. Hassan

We develop a basis-resolved covariance framework for investigating the organization of low-energy many-body wavefunctions in the spin- $1/2$ Kitaev--Heisenberg model. By constructing covariance matrices from local-spin, bond-correlation, and plaquette-flux representations of the low-energy states, Principal Component Analysis (PCA) is employed to identify the dominant collective covariance modes. We find a systematic evolution of the covariance geometry across the phase diagram: magnetically ordered phases are described by an essentially one-dimensional covariance manifold, conventional magnetic phase boundaries exhibit competition between leading covariance modes, whereas the Kitaev regimes develop intrinsically multidimensional covariance geometry. Furthermore, the same many-body wavefunction produces distinct covariance geometries in different operator representations, demonstrating that covariance geometry is determined jointly by the quantum state and the physical observables used to probe it. Shannon entropy and the participation ratio provide quantitative measures of this evolution. The present framework establishes basis-resolved covariance geometry as a complementary statistical perspective for characterizing frustrated quantum many-body systems beyond conventional order parameters.We develop a basis-resolved covariance framework for investigating the organization of low-energy many-body wavefunctions in the spin-$1/2$ Kitaev--Heisenberg model. By constructing covariance matrices from local-spin, bond-correlation, and plaquette-flux representations of the low-energy states, Principal Component Analysis (PCA) is employed to identify the dominant collective covariance modes. We find a systematic evolution of the covariance geometry across the phase diagram: magnetically ordered phases are described by an essentially one-dimensional covariance manifold, conventional magnetic phase boundaries exhibit competition between leading covariance modes, whereas the Kitaev regimes develop intrinsically multidimensional covariance geometry. Furthermore, the same many-body wavefunction produces distinct covariance geometries in different operator representations, demonstrating that covariance geometry is determined jointly by the quantum state and the physical observables used to probe it. Shannon entropy and the participation ratio provide quantitative measures of this evolution. The present framework establishes basis-resolved covariance geometry as a complementary statistical perspective for characterizing frustrated quantum many-body systems beyond conventional order parameters.

arXiv:2607.25462Jul 28, 20260 citations

Invariance Principles

P. K. Kabir

Springer eBooksJan 1, 19762 citations

n

), and their associated Cartan involution provides a route to exactly compute the eigenspectrum of their Choi states and the partial transpose for any

n\in \mathbb{N}

. These channels include previously studied SO

(n)

-covariant channels, and further include Sp

(\frac{n}{2})

-covariant channels (when

n

is even) and S(U(

p

)

\times

q

))-covariant channels (where

n\!=\!p\!+\!q

). We show that all Cartan-covariant channels satisfy the PPT

^{2}

-conjecture and further demonstrate the nontrivial nature of this result for the class of Sp

(\frac{n}{2})

-covariant and

{\rm S}({\rm U}(p) \!\times\!{\rm U}(q))

-covariant channels

Covariance Geometry of Basis-Resolved Low-Energy Wavefunctions in the Spin-

1/2

Kitaev--Heisenberg Model

We develop a basis-resolved covariance framework for investigating the organization of low-energy many-body wavefunctions in the spin-

1/2

Kitaev--Heisenberg model. By constructing covariance matrices from local-spin, bond-correlation, and plaquette-flux representations of the low-energy states, Principal Component Analysis (PCA) is employed to identify the dominant collective covariance modes. We find a systematic evolution of the covariance geometry across the phase diagram: magnetically ordered phases are described by an essentially one-dimensional covariance manifold, conventional magnetic phase boundaries exhibit competition between leading covariance modes, whereas the Kitaev regimes develop intrinsically multidimensional covariance geometry. Furthermore, the same many-body wavefunction produces distinct covariance geometries in different operator representations, demonstrating that covariance geometry is determined jointly by the quantum state and the physical observables used to probe it. Shannon entropy and the participation ratio provide quantitative measures of this evolution. The present framework establishes basis-resolved covariance geometry as a complementary statistical perspective for characterizing frustrated quantum many-body systems beyond conventional order parameters.